log2N

Log2n refers to the logarithm of a positive number n with base 2. It is written as log base 2 of n, or log2 n, and is the exponent x that satisfies 2^x = n. Like other logarithms, log2 n is the inverse of the exponential function 2^x and is defined for n > 0. If n is a power of two, log2 n is an integer; otherwise it is a non-integer real number.

Key properties include that log2 n is strictly increasing on (0, infinity) and maps (0, infinity) to (-infinity, infinity). Its derivative with respect to n is 1/(n ln 2). The identity log2(2^k) = k holds for any real k. The change of base formula gives log2 n = log n / log 2 for any other base of the logarithm. For integer n ≥ 1, floor(log2 n) is the position of the highest set bit in the binary representation, and the bit-length of n is floor(log2 n) + 1.

In practice, log2 n is widely used in computer science to analyze algorithmic complexity, often appearing in time bounds that are O(log n). It also relates to the amount of information in binary systems, where logarithms base 2 measure information content. When applying log2 n to discrete contexts, rounding is common; for example, floor or ceil functions are used to obtain integer results from non-integer logarithms.